Bernstein-Schnabl operators: new achievements and perspectives

Bernstein-Schnabl operators were first introduced by R. Schnabl in 1968 in the context of the sets of all probability Radon measures on compact Hausdorff spaces, in order to present a unitary treatment of the saturation problem for Bernstein operators on the unit interval and on the finite dimensional simplices and hypercubes. These operators were referred to as Bernstein-Schnabl operators in 1971 by G. Felbecker and W. Schempp who extended their definition by considering arbitrary infinite lower triangular stochastic matrices. Finally, in 1974 M. W. Grossman systematized the same definition in the more appropriate setting of (not necessarily finite dimensional) convex compact subsets. In 1989 the present author undertook a detailed study of these operators in the special case where they are linked to a positive (linear) projection, showing a deep interplay between them, the theory of Feller semigroups and the relevant initial-boundary value differential problems and their probabilistic counterparts. The main aspects of this theory, which has been also accomplished by other authors, is documented in Chapter 6 of [1]. Motivated by the need to enlarge the classes of both approximation and differential problems where a similar theory could be applied, recently a joint project with M. Cappelletti Montano, V. Leonessa and I. Raşa has been undertaken in order to extend the above mentioned theory by considering arbitrary positive (linear) operators instead of positive projections. This new challenging approach has disclosed new and more difficult problems. The talk will be centered about some results obtained with this new approach whose details will appear in the forthcoming monograph [2].